Could the ancient Romans have built a digital computer?

It's an alternate history; the point of departure being that Archimedes of Syracuse was captured, taken to Rome and then funded (because the Romans were nothing if not practical). In her setting, the Romans are well into an industrial revolution.

The blurb she wrote for an agent:

    Pontius Pilate is a successful corporate lawyer 
    headhunted by the Roman Senate to run a difficult 
    province in need of a gentler, civilian touch. He’s 
    been in the job five years — and is starting to get 
    the hang of it — when the Yeshua Ben Yusuf file 
    lands on his desk. This is not ideal right now. 
    Judaea is in the midst of a major terrorism crisis, 
    his wife keeps threatening to go back to Caesarea (
    she can’t stand Jerusalem), his son is becoming far 
    too friendly with the High Priest’s son, and his 
    boss keeps forgetting that he isn’t actually in the 
    army. Even worse, his closest friend and greatest 
    rival from law school is Ben Yusuf’s lawyer. A 
    Jerusalem courtroom is the setting for the first 
    clash of civilisations, where people from a 
    fundamentally different tradition are forced to 
    engage with religious ideas that in many respects 
    they do not want to understand.

    What is most distinctive about the book is my 
    imagining of what a technologically advanced pagan 
    civilisation would look like. That is, what if the 
    Romans won much of their empire under conditions 
    that we associate with the Industrial Revolution? 
    What if — with their distinctive, non-Christian 
    moral values — they were gifted with all that 
    immense fire-power and confronted with monotheistic 
    terrorists?
    
    I do not think the Romans were secular in the 
    modern sense, and I haven’t portrayed them as such. 
    They were, however, very different from the 
    monotheistic peoples they confronted in Judaea. My 
    Roman characters are still religious, but 
    differently religious. Unlike many authors of a 
    skeptical bent, I do not seek to score cheap shots 
    by denigrating religion per se. Rather, Bring Laws 
    & Gods recognises the persistent vitality of 
    religious traditions, especially when their 
    practitioners are confronted with overwhelming 
    military power and physical occupation by non-
    believers.

http://skepticlawyer.com.au/2010/09/22/the-past-is-a-foreign...

http://skepticlawyer.com.au/2010/09/24/the-angel-bring-laws-... (NSFW)

http://skepticlawyer.com.au/2010/10/15/the-visit-bring-laws-...

http://skepticlawyer.com.au/2010/11/25/patria-patria/ (NSFW)

Not exact models of what a modernist Rome would look like, though. For one thing the Roman legal system, in its various forms, was a different creature. They had slavery ... but also very liberal marriage and divorce laws compared to most societies in history.

They're also one of only two legal systems which developed torts -- in Roman law, delicts. As an innovation torts/delicts take a lot of pressure off the state to provide regulation of commercial and personal behaviour, because a lot of it is then handled at the lower level by individuals and companies.